`f(x) = sin(x), [0,2pi]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open...

`f(x) = sin(x), [0,2pi]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open interval

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Chapter 3, 3.2 - Problem 17 - Calculus of a Single Variable (10th Edition, Ron Larson).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Yes, it can. Function f is continuous on `[0, 2pi]` and is differentiable on` (0, 2pi)` . Also, `f(0) = f(2pi)` (both =0). There are all conditions of Rolle's Theorem.

Because of this there is at least one point c in `[0, 2pi]` such that f'(c)=0.

f'(x) = cos(x), it is zero at two points from `[0, 2pi]` : `pi/2` and `(3pi)/2` .

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